A mathematical model for value estimation with public information and herding

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March 2014, 7(1): 29-44. doi: 10.3934/krm.2014.7.29

A mathematical model for value estimation with public information and herding

Marcello Delitala ^1, and Tommaso Lorenzi ^1,

1.	Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received November 2012 Revised April 2013 Published December 2013

Abstract
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This paper deals with a class of integro-differential equations modeling the dynamics of a market where agents estimate the value of a given traded good. Two basic mechanisms are assumed to concur in value estimation: interactions between agents and sources of public information and herding phenomena. A general well-posedness result is established for the initial value problem linked to the model and the asymptotic behavior in time of the related solution is characterized for some general parameter settings, which mimic different economic scenarios. Analytical results are illustrated by means of numerical simulations and lead us to conclude that, in spite of its oversimplified nature, this model is able to reproduce some emerging behaviors proper of the system under consideration. In particular, consistently with experimental evidence, the obtained results suggest that if agents are highly confident in the product, imitative and scarcely rational behaviors may lead to an over-exponential rise of the value estimated by the market, paving the way to the formation of economic bubbles.

Keywords: integro-differential equations., asymptotic analysis, Socio-economic systems, weak convergence, emerging behaviors.

Mathematics Subject Classification: Primary: 35R09, 91D10; Secondary: 35B4.

Citation: Marcello Delitala, Tommaso Lorenzi. A mathematical model for value estimation with public information and herding. Kinetic & Related Models, 2014, 7 (1) : 29-44. doi: 10.3934/krm.2014.7.29

References:

[1]	G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economic systems by functional subsystems representation,, Kin. Rel. Mod., 1 (2008), 249. doi: 10.3934/krm.2008.1.249. Google Scholar
[2]	P. Ball, Why Society is a Complex Matter,, Springer-Verlag, (2012). doi: 10.1007/978-3-642-29000-8. Google Scholar
[3]	J. Berg, M. Marsili, A. Rustichini and R. Zecchina, Statistical mechanics of asset markets with private information,, J. Quant. Finance, 1 (2001), 203. Google Scholar
[4]	Y. Biondi, P. Giannoccolo and S. Galam, Formation of share market prices under heterogeneous beliefs and common knowledge,, Physica A, 391* (2012), 5532. doi: 10.1016/j.physa.2012.06.015. Google Scholar*
[5]	A. Boccabella, R. Natalini and L. Pareschi, On a continuous mixed strategy model for evolutionary game-theory,, Kinet. Relat. Models, 4 (2011), 187. doi: 10.3934/krm.2011.4.187. Google Scholar
[6]	D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence,, Commun. Pure Appl. Anal., 12 (2013), 1487. doi: 10.3934/cpaa.2013.12.1487. Google Scholar
[7]	J.-P. Bouchaud, Economics need a scientific revolution,, Nature, 455 (2008). doi: 10.1038/4551181a. Google Scholar
[8]	J.-P. Bouchaud, The (unfortunate) complexity of the economy,, Physics World, 82 (2009), 28. Google Scholar
[9]	L. Boudin and F. Salvarani, Modeling Opinion Formation by Means of Kinetic Equations,, in Mathematical modeling of collective behavior in socio-economic and life sciences, (2010). doi: 10.1007/978-0-8176-4946-3_10. Google Scholar
[10]	L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior,, Kinet. Relat. Models, 2 (2009), 433. doi: 10.3934/krm.2009.2.433. Google Scholar
[11]	S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy,, J. Stat. Phys., 120 (2005), 253. doi: 10.1007/s10955-005-5456-0. Google Scholar
[12]	M. Cristelli, L. Pietronero and A. Zaccaria, Critical Overview of Agent-Based Models for Economics,, Proceedings of the School of Physics E. Fermi, (2010). Google Scholar
[13]	B. During, P. A. Markowich, J. F. Pietschmann and M. T. Wolfram, Boltzmann and Fokker-Planck Equations modelling Opinion Formation in the Presence of strong Leaders,, Proc. Royal Soc. A, 465 (2009), 3687. doi: 10.1098/rspa.2009.0239. Google Scholar
[14]	R. Gatignol, Théorie Cinétique des Gaz à Répartition Discréte des Vitèsses,, (French) Lecture Notes in Physics, (1975). Google Scholar
[15]	E. Eyster and M. Rabin, naïve herding in rich-information settings,, AEJ Microeconomics, 2 (2010), 221. doi: 10.1257/mic.2.4.221. Google Scholar
[16]	T. Kaizoji and D. Sornette, Market Bubbles and Crashes,, in Encyclopedia of quantitative finance, (2010). doi: 10.1143/PTPS.162.165. Google Scholar
[17]	J.-M. Lasry and P.-L. Lions, Mean field games,, Japan. J. Math., 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. Google Scholar
[18]	J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey,, International Journal of Modern Physics C, 18 (2007), 1819. doi: 10.1142/S0129183107011789. Google Scholar
[19]	T. Lux, Herd behaviour, bubbles and crashes,, Economic Journal, 105 (1995), 881. doi: 10.2307/2235156. Google Scholar
[20]	D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets,, Physica A, 391 (2012), 715. doi: 10.1016/j.physa.2011.08.013. Google Scholar
[21]	M. Marsili, D. Challet and R. Zecchina, Exact solution of a modified El Farol's bar problem: Efficiency and the role of market impact,, Physica A: Statistical Mechanics and its Applications, 280* (2000), 522. doi: 10.1016/S0378-4371(99)00610-X. Google Scholar*
[22]	Q. Michard and J.-P. Bouchaud, Theory of collective opinion shifts: From smooth trends to abrupt swings,, Eur. Phys. J. B, 47 (2005), 151. doi: 10.1140/epjb/e2005-00307-0. Google Scholar
[23]	B. Perthame, Trasport Equations in Biology,, Frontiers in Mathematics. Birkäuser Verlag, (2007). Google Scholar
[24]	H. A. Simon, Invariants of human behavior,, Annu. Rev. Psychol., 41 (1990), 1. Google Scholar
[25]	G. Toscani, C. Brugna and S. Demichelis, Kinetic models for the trading of goods,, J. Stat. Phys., 151* (2013), 549. doi: 10.1007/s10955-012-0653-0. Google Scholar*

show all references

References:

[1]	G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economic systems by functional subsystems representation,, Kin. Rel. Mod., 1 (2008), 249. doi: 10.3934/krm.2008.1.249. Google Scholar
[2]	P. Ball, Why Society is a Complex Matter,, Springer-Verlag, (2012). doi: 10.1007/978-3-642-29000-8. Google Scholar
[3]	J. Berg, M. Marsili, A. Rustichini and R. Zecchina, Statistical mechanics of asset markets with private information,, J. Quant. Finance, 1 (2001), 203. Google Scholar
[4]	Y. Biondi, P. Giannoccolo and S. Galam, Formation of share market prices under heterogeneous beliefs and common knowledge,, Physica A, 391* (2012), 5532. doi: 10.1016/j.physa.2012.06.015. Google Scholar*
[5]	A. Boccabella, R. Natalini and L. Pareschi, On a continuous mixed strategy model for evolutionary game-theory,, Kinet. Relat. Models, 4 (2011), 187. doi: 10.3934/krm.2011.4.187. Google Scholar
[6]	D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence,, Commun. Pure Appl. Anal., 12 (2013), 1487. doi: 10.3934/cpaa.2013.12.1487. Google Scholar
[7]	J.-P. Bouchaud, Economics need a scientific revolution,, Nature, 455 (2008). doi: 10.1038/4551181a. Google Scholar
[8]	J.-P. Bouchaud, The (unfortunate) complexity of the economy,, Physics World, 82 (2009), 28. Google Scholar
[9]	L. Boudin and F. Salvarani, Modeling Opinion Formation by Means of Kinetic Equations,, in Mathematical modeling of collective behavior in socio-economic and life sciences, (2010). doi: 10.1007/978-0-8176-4946-3_10. Google Scholar
[10]	L. Boudin and F. Salvarani, The quasi-invariant limit for a kinetic model of sociological collective behavior,, Kinet. Relat. Models, 2 (2009), 433. doi: 10.3934/krm.2009.2.433. Google Scholar
[11]	S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy,, J. Stat. Phys., 120 (2005), 253. doi: 10.1007/s10955-005-5456-0. Google Scholar
[12]	M. Cristelli, L. Pietronero and A. Zaccaria, Critical Overview of Agent-Based Models for Economics,, Proceedings of the School of Physics E. Fermi, (2010). Google Scholar
[13]	B. During, P. A. Markowich, J. F. Pietschmann and M. T. Wolfram, Boltzmann and Fokker-Planck Equations modelling Opinion Formation in the Presence of strong Leaders,, Proc. Royal Soc. A, 465 (2009), 3687. doi: 10.1098/rspa.2009.0239. Google Scholar
[14]	R. Gatignol, Théorie Cinétique des Gaz à Répartition Discréte des Vitèsses,, (French) Lecture Notes in Physics, (1975). Google Scholar
[15]	E. Eyster and M. Rabin, naïve herding in rich-information settings,, AEJ Microeconomics, 2 (2010), 221. doi: 10.1257/mic.2.4.221. Google Scholar
[16]	T. Kaizoji and D. Sornette, Market Bubbles and Crashes,, in Encyclopedia of quantitative finance, (2010). doi: 10.1143/PTPS.162.165. Google Scholar
[17]	J.-M. Lasry and P.-L. Lions, Mean field games,, Japan. J. Math., 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. Google Scholar
[18]	J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey,, International Journal of Modern Physics C, 18 (2007), 1819. doi: 10.1142/S0129183107011789. Google Scholar
[19]	T. Lux, Herd behaviour, bubbles and crashes,, Economic Journal, 105 (1995), 881. doi: 10.2307/2235156. Google Scholar
[20]	D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets,, Physica A, 391 (2012), 715. doi: 10.1016/j.physa.2011.08.013. Google Scholar
[21]	M. Marsili, D. Challet and R. Zecchina, Exact solution of a modified El Farol's bar problem: Efficiency and the role of market impact,, Physica A: Statistical Mechanics and its Applications, 280* (2000), 522. doi: 10.1016/S0378-4371(99)00610-X. Google Scholar*
[22]	Q. Michard and J.-P. Bouchaud, Theory of collective opinion shifts: From smooth trends to abrupt swings,, Eur. Phys. J. B, 47 (2005), 151. doi: 10.1140/epjb/e2005-00307-0. Google Scholar
[23]	B. Perthame, Trasport Equations in Biology,, Frontiers in Mathematics. Birkäuser Verlag, (2007). Google Scholar
[24]	H. A. Simon, Invariants of human behavior,, Annu. Rev. Psychol., 41 (1990), 1. Google Scholar
[25]	G. Toscani, C. Brugna and S. Demichelis, Kinetic models for the trading of goods,, J. Stat. Phys., 151* (2013), 549. doi: 10.1007/s10955-012-0653-0. Google Scholar*

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